The deficiency of a propositional formula F in CNF with n variablesand m clauses is defined as m-n. It is known that minimalunsatisfiable formulas (unsatisfiable formulas which becomesatisfiable by removing any clause) have positive deficiency.Recognition of minimal unsatisfiable formulas is NP-hard, and it wasshown recently ...
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Bayesian inference and counting satisfying assignments are importantproblems with numerous applications in probabilistic reasoning. In thispaper, we show that plain old DPLL equipped with memoization can solveboth of these problems with time complexity that is at least asgood as all known algorithms. Furthermore, DPLL with memoizationmore >>>

We present a simple proof of the bounded-depth Frege lower bounds ofPitassi et. al. and Krajicek et. al. for the pigeonholeprinciple. Our method uses the interpretation of proofs as two playergames given by Pudlak and Buss. Our lower bound is conceptuallysimpler than previous ones, and relies ...
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Given a Boolean function f, we study two natural generalizations of the certificate complexity C(f): the randomized certificate complexity RC(f) and the quantum certificate complexity QC(f). Using Ambainis' adversary method, we exactly characterize QC(f) as the square root of RC(f). We then use this result to prove the new relation ...
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For a boolean formula \phi on n variables, the associated property P_\phi is the collection of n-bit strings that satisfy \phi. We provethat there are 3CNF properties that require a linear number of queries,even for adaptive tests. This contrasts with 2CNF propertiesthat are testable with O(\sqrt{n}) ...
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$k$-SAT is one of the best known among a wide class of random constraint satisfaction problems believed to exhibit a threshold phenomenon where the control parameter is the ratio, number of constraints to number of variables. There has been a large amount of work towards estimating ...
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We study the role of connectivity of communication networks in privatecomputations under information theoretic settings. It will be shown thatsome functions can be computed by private protocols even if theunderlying network is 1-connected but not 2-connected. Then we give acomplete characterisation of non-degenerate functions that can ...
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The satisfiability problem of Boolean Formulae in 3-CNF (3-SAT) is a well known NP-complete problem and the development of faster(moderately exponential time) algorithms has received much interest in recent years. We show that the 3-SAT problem can be solved by aprobabilistic algorithm in expected time O(1,3290^n).more >>>

We study the question of whether the class DisNP of disjoint pairs (A, B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists ...
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We prove that the Cutting Plane proof system based onGomory-Chvatal cuts polynomially simulates the lift-and-project system with integer coefficientswritten in unary. The restriction on coefficients can beomitted when using Krajicek's cut-free Gentzen-style extensionof both systems. We also prove that Tseitin tautologieshave short proofs in ...
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Cryan and Miltersen recently considered the question of whether there can be a pseudorandom generator in NC0, that is, a pseudorandom generator such that every bit of the output depends on a constant number k of bits of the seed. They show that for k=3 there ...
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We introduce a ``Statistical Query Sampling'' model, in which the goal of an algorithm is to produce an element in a hidden set$S\subseteq\bit^n$ with reasonable probability. The algorithmgains information about $S$ through oracle calls (statisticalqueries), where the algorithm submits a query function $g(\cdot)$ and receives ...
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In 1986, Fiat and Shamir suggested a general method for transforming secure 3-round public-coin identification schemes into digital signature schemes. The significant contribution of this method is a means for designing efficient digital signatures, while hopefully achieving security against chosen message attacks. All other known constructions which achieve such security ...
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In this work, we study the stability of the {\sf FIFO} ({\sfFirst-In-First-Out}) protocol in the context of AdversarialQueueing Theory. As an important intermediate step, we consider{\em dynamic capacities}, where each network link capacity mayarbitrarily take on values in the two-valued set of integers$\{1,C\}$ for $C>1$ ...
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The best-known representations of boolean functions f are those of disjunctions of terms (DNFs) and as conjuctions of clauses (CNFs). It is convenient to define the DNF size of f as the minimal number of terms in a DNF representing f and the CNF size as the minimal number of ...
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We show that the class of integer-valued functions computable bypolynomial-space Turing machines is exactly the class of functions ffor which there is a nondeterministic polynomial-time Turingmachine with a certain order on its paths that on input x outputs a 3x3matrix with entries from {-1,0,1} on each ...
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We present upper bounds on the size of codes that are locallytestable by querying only two input symbols. For linear codes, weshow that any $2$-locally testable code with minimal distance$\delta n$ over a finite field $F$ cannot have more than$|F|^{3/\delta}$ codewords. This result holds even ...
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We consider possible equality QMA=PP and give an argumentagainst it. Namely, this equality implies that PP contains PH. The argument is based on the strong form of Toda's theorem andthe strengthening of the proof for inclusion $QMA\subseteq PP$ due to Kitaev and Watrous.

We study approximation hardness and satisfiability of bounded occurrence uniform instances of SAT. Among other things, we prove the inapproximability for SAT instances in which every clause has exactly 3 literals and each variable occurs exactly 4 times, and display an explicit ...
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In an attempt to generalize Christofides algorithm for metric TSP to the asymmetric TSP with triangle inequality we have studied various properties of directed spanning cacti. In this paper we first observe that finding the TSP in a directed, weighted complete graph with triangle inequality is polynomial time equivalent to ...
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Kummer's cardinality theorem states that a language is recursiveif a Turing machine can exclude for any n words one of then + 1 possibilities for the number of words in the language. It is known that this theorem does not hold for polynomial-time computations, but there ...
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We obtain a characterization of ACC0 in terms of a natural class ofconstant width circuits, namely in terms of constant width polynomialsize planar circuits. This is shown via a characterization of theclass of acyclic digraphs which can be embedded on a cylinder surfacein such a way ...
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We study small degree graph problems such as Maximum Independent Setand Minimum Node Cover and improve approximation lower bounds for them and for a number of related problems, like Max-B-Set Packing, Min-B-Set Cover, Max-Matching in B-uniform 2-regular hypergraphs. For example, we prove NP-hardness factor of 95/94more >>>

We prove that all of the following assertions are equivalent:There is a many-one complete disjoint NP-pair; there is a strongly many-one complete disjoint NP-pair;there is a Turing complete disjoint NP-pair such that all reductions are smart reductions;there is a complete disjoint NP-pair for one-to-one, invertible ...
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An almost complete set A for a complexity class C is a language of C which is not complete, but that has the property that ``many'' languages of C reduce to A, where the term ``many'' is used in reference to Lutz's resource bounded measure (rbm). The question of the ...
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We prove that BPP has Lutz's p-dimension at most 1/2 unless BPP equals EXP. Next we show that BPP has Lutz's p-dimension zero unless BPP equals EXP on infinitely many input lengths. We also prove that BPP has measure zero in the smaller complexity class ...
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Abstract. It is known that random k-SAT formulas with at least(2^k*ln2)*n random clauses are unsatisfiable with high probability. Thisresult is simply obtained by bounding the expected number of satisfy-ing assignments of a random k-SAT instance by an expression tendingto 0 when n, the number of variables ...
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Both average-case complexity and the study of the approximability properties of NP-optimization problems are well established and active fields of research. By applying the notion of average-case complexity to approximation problems we provide a formal framework that allows the classification of NP-optimization problems according to their average-case approximability. Thus, known ...
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We investigate the hardness of approximating the longest path and the longest cycle in directed graphs on $n$ vertices. We show that neither of these two problems can be polynomial time approximated within $n^{1-\epsilon}$ for any $\epsilon>0$ unless $\text{P}=\text{NP}$. In particular, the result holds formore >>>

Traditionally, communication networks are composed ofrouting nodes, which relay and duplicate data. Work inrecent years has shown that for the case of multicast, animprovement in both rate and code-construction complexity can begained by replacing these routing nodes by linear codingnodes. These nodes transmit linear combinations ...
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Height restricted constant depth LK is a natural restriction of Gentzen's propositional proof system LK. A sequence of LK-formulas has polylogarithmic-height restricted depth-d-LK proofs iff the n-th formula in the sequence possesses LK-proofs where cut-formulas are of depth d+1 with small bottom fanin and of ...
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In the {\sc $k$-center} problem, the input is a bound $k$ and $n$ points with the distance between every two of them,such that the distances obey the triangle inequality.The goal is to choose a set of $k$ points to serve as centers,so that the maximum distance ...
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The eliminationproblem is classical:implicitly express one of the variables occurring in a finitesystem of polynomial equations as an algebraic function of adesignated subset of the remaining variables. Solutions to thisproblem by resultants, or more comprehensively byuse of Gr\"{o}bner basis methods are available. In this ...
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We present a novel technique, based on the Jensen-Shannon divergencefrom information theory, to prove lower bounds on the query complexityof sampling algorithms that approximate functions over arbitrarydomain and range. Unlike previous methods, our technique does not use a reduction from a binary decision problem, but rather ...
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We show that the asymmetric $k$-center problem is$\Omega(\log^* n)$-hard to approximate unless${\rm NP} \subseteq {\rm DTIME}(n^{poly(\log \log n)})$.Since an $O(\log^* n)$-approximation algorithm is knownfor this problem, this essentially resolves the approximabilityof this problem. This is the first natural problemwhose approximability threshold does not polynomially ...
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A theory, in this context, is a Boolean formula; it isused to classify instances, or truth assignments. Theoriescan model real-world phenomena, and can do so more or lesscorrectly. The theory revision, or concept revision, problem is tocorrect a given, roughly correct concept.This problem is ...
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We use recent results on the hardness of resource-bounded Kolmogorov random strings, to prove that RP is small in SUBEXP else ZPP=PSPACE and NP=EXP. We also prove that if NP is not small in SUBEXP, then NP=AM, improving a former result which held for the measure ...
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We study the Chv\'atal rank of polytopes as a complexity measure ofunsatisfiable sets of clauses. Our first result establishes aconnection between the Chv\'atal rank and the minimum refutationlength in the cutting planes proof system. The result implies thatlength lower bounds for cutting planes, or even for ...
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We show that Yao's XOR Lemma, and its essentially equivalentrephrasing as a Direct Product Lemma, can bere-interpreted as a way of obtaining error-correctingcodes with good list-decoding algorithms from error-correctingcodes having weak unique-decoding algorithms. To get codeswith good rate and efficient list decoding algorithmsone needs ...
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Cryan and Miltersen recently considered the questionof whether there can be a pseudorandom generator inNC0, that is, a pseudorandom generator such that everybit of the output depends on a constant number k of bitsof the seed. They show that for k=3 there is always adistinguisher; ...
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We show that the Player-Adversary game from a paperby Pudlak and Impagliazzo played overCNF propositional formulas gives an exact characterization of the space neededin treelike resolution refutations. Thischaracterization is purely combinatorial and independent of the notion of resolution.We use this characterization to give ...
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We initiate a general study of pseudo-random implementationsof huge random objects, and apply it to a few areasin which random objects occur naturally.For example, a random object being considered may bea random connected graph, a random bounded-degree graph,or a random error-correcting code with good ...
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We strengthen an earlier notion ofresource-bounded Baire's categories, and defineresource bounded Baire's categories on small complexity classes such as P, QP, SUBEXPand on probabilistic complexity classes such as BPP.We give an alternative characterization of meager sets via resource-boundedBanach Mazur games.We show that the class ...
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We study the problem of representing symmetric Boolean functions as symmetric polynomials over Z_m. We show an equivalence between suchrepresentations and simultaneous communication protocols. Computing a function with a polynomial of degree d modulo m=pq is equivalent to a two player protocol where one player is given the first ...
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Randomized search heuristics like local search, simulated annealing or all kinds of evolutionary algorithms have many applications. However, for most problems the best worst-case expected run times are achieved by more problem-specific algorithms. This raises the question about the limits of general randomized search heuristics.

We prove approximation hardness of short symmetric instances of MAX-3SAT in which each literal occurs exactly twice, and each clause is exactly of size 3. We display also an explicit approximation lower bound for that problem. The bound two on the number ...
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A CNF formula is k-satisfiable if each k clauses of it can be satisfiedsimultaneously. Let \pi_k be the largest real number such that for eachk-satisfiable formula with variables x_i, there are probabilities p_iwith the following property: If each variable x_i is chosen randomly andindependently to be ...
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Johnson, Papadimitriou and Yannakakis introduce the class $\PLS$ consisting of optimization problems for which efficient local-search heuristics exist. We formulate a type-2 problem $\iter$that characterizes $\PLS$ in style of Beame et al., and provea criterion for type-2 problems to be nonreducible to $\iter$.As a corollary, ...
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We show that for a graph G it is NP-hard to decide whether its independence number alpha(G) equals its clique partition number ~chi(G) even when some minimum clique partition of G is given. This implies that any alpha(G)-upper bound provably better than ~chi(G) is NP-hard to compute.

This paper presents a new upper bound for the$k$-satisfiability problem. For small $k$'s, especially for $k=3$,there have been a lot of algorithms which run significantly fasterthan the trivial $2^n$ bound. The following list summarizes thosealgorithms where a constant $c$ means that the algorithm runs in timemore >>>

This paper establishes a randomized algorithm that finds a satisfying assignment for a satisfiable formula $F$ in 3-CNF in $O(1.32793^n)$ expected running time. The algorithms is based on the analysis of so-called strings, which are sequences of 3-clauses where non-succeeding clauses do not share a variable and succeeding clauses share ...
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We describe a general method how to construct froma propositional proof system P a possibly much strongerproof system iP. The system iP operates withexponentially long P-proofs described ``implicitly''by polynomial size circuits.

As an example we prove that proof system iEF, implicit EF,corresponds to bounded ...
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We prove that the problems of minimum bisection on k-uniform hypergraphs are almost exactly as hard to approximate, up to the factor k/3, as the problem of minimum bisection on graphs. On a positive side, our argument gives also the first approximation ...
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The problem of finding a local minimum of a black-box function is centralfor understanding local search as well as quantum adiabatic algorithms.For functions on the Boolean hypercube {0,1}^n, we show a lower bound ofOmega(2^{n/4}/n) on the number of queries needed by a quantum computer tosolve this ...
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We show that a certain representation of the matrix-product can be computed with $n^{o(1)}$ multiplications. We also show, that similar representations of matrices can be compressed enormously with the help of simple linear transforms.

Advice is supplementary information that enhances the computationalpower of an underlying computation. This paper focuses on advice thatis given in the form of a pure quantum state. The notion of advisedquantum computation has a direct connection to non-uniform quantumcircuits and tally languages. The paper examines the ...
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A Secure Function Evaluation (SFE) of a two-variable function f(.,.) is a protocol that allows two parties with inputs x and y to evaluate f(x,y) in a manner where neither party learns ``more than is necessary". A rich body of work deals with the study of completeness for secure ...
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Free binary decision diagrams (FBDDs) are graph-based data structures representing Boolean functions with a constraint (additional to binary decision diagrams) that each variable is tested at most once during the computation. The function EAR_n is the following Boolean function definedfor n x n Boolean matrices: EAR_n(M)=1 iff the matrix ...
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In constraint satisfaction problems over finite domains, some variablescan be frozen, that is, they take the same value in all possible solutions. We study the complexity of the problem of recognizing frozen variables with restricted sets of constraint relations allowed in theinstances. We show that the complexity of ...
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Derandomization techniques are used to show that at least one of thefollowing holds regarding the size of the counting complexity classSPP.1. SPP has p-measure 0.2. PH is contained in SPP.In other words, SPP is small by being a negligible subset ofexponential time or large ...
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Given a polynomial f(X) with rational coefficients as inputwe study the problem of (a) finding the order of the Galois group off(X), and (b) determining the Galois group of f(X) by finding a smallgenerator set. Assuming the generalized Riemann hypothesis, we provethe following complexity bounds:

A source is compressible if we can efficiently compute shortdescriptions of strings in the support and efficientlyrecover the strings from the descriptions. In this paper, wepresent a technique for proving lower bounds oncompressibility in an oracle setting, which yields thefollowing results:

Lattices have received considerable attention as a potential source of computational hardness to be used in cryptography, after a breakthrough result of Ajtai (STOC 1996) connecting the average-case and worst-case complexity of various lattice problems. The purpose of this paper is twofold. On the expository side, we present a rigorous ...
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An arithmetic formula is multi-linear if the polynomial computedby each of its sub-formulas is multi-linear. We prove that anymulti-linear arithmetic formula for the permanent or thedeterminant of an $n \times n$ matrix is of size super-polynomialin $n$.

SBP is a probabilistic promise class locatedbetween MA and AM \cap BPPpath. The firstpart of the paper studies the question of whetherSBP has many-one complete sets. We relatethis question to the existence of uniformenumerations. We construct an oracle relative towhich SBP and AM do ...
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We consider the approximate nearest neighbour search problem on theHamming Cube $\b^d$. We show that a randomised cell probe algorithm thatuses polynomial storage and word size $d^{O(1)}$ requires a worst casequery time of $\Omega(\log\log d/\log\log\log d)$. The approximationfactor may be as loose as $2^{\log^{1-\eta}d}$ for any ...
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We study private computations in information-theoretical settings onnetworks that are not 2-connected. Non-2-connected networks are``non-private'' in the sense that most functions cannot privately becomputed on such networks. We relax the notion of privacy byintroducing lossy private protocols, which generalize privateprotocols. We measure the information each ...
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We present a simple randomized algorithm for SAT and prove an upper bound on its running time. Given a Boolean formula F in conjunctive normal form, the algorithm finds a satisfying assignment for F(if any) by repeating the following: Choose an assignment A at random and ...
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We study the Lovasz number theta along with two further SDP relaxations $\thetI$, $\thetII$of the independence number and the corresponding relaxations of thechromatic number on random graphs G(n,p). We prove that \theta isconcentrated about its mean, and that the relaxations of the chromaticnumber in the case ...
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We consider the following phenomenon: with just one multiplication we can compute (3u+2v)(3x+2y)= 3ux+4vy mod 6, while computing the same polynomial modulo 5 needs 2 multiplications. We generalize this observation and we define some vectors, called sixtors, with remarkable zero-divisor properties. Using sixtors, we also generalize our earlier result ...
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Communication complexity is concerned with the question: how much information do the participants of a communication system need to exchange in order to perform certain tasks? The minimum number of bits that must be communicated is the deterministic communication complexity of $f$. This complexity measure was introduced by Yao \cite{1} ...
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A $k$-uniform hypergraph $G$ of size $n$ is said to be $\varepsilon$-far from having an independent set of size $\rho n$ if one must remove at least $\varepsilon n^k$ edges of $G$ in order for the remaining hypergraph to have an independent set of size $\rho n$. In this work, ...
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This paper introduces logspace optimisation problems as analogues of the well-studied polynomial-time optimisationproblems. Similarly to them, logspaceoptimisation problems can have vastly different approximationproperties, even though the underlying existence and budget problemshave the same computational complexity. Numerous natural problemsare presented that exhibit such a varying ...
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We investigate a new type of information-theoretic identification problem, suggested to us by Alan Taylor. In this problem we are given a set of items, more than half of which share a common ``good" value. The other items have various other values which are called ``bad". The only method we ...
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Several researchers, including Leonid Levin, Gerard 't Hooft, and Stephen Wolfram, have argued that quantum mechanics will break down before the factoring of large numbers becomes possible. If this is true, then there should be a natural "Sure/Shor separator" -- that is, a set of quantum ...
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We present an explicit construction of codes that can be list decodedfrom a fraction $(1-\eps)$ of errors in sub-exponential time and whichhave rate $\eps/\log^{O(1)}(1/\eps)$. This comes close to the optimalrate of $\Omega(\eps)$, and is the first sub-exponential complexityconstruction to beat the rate of $O(\eps^2)$ achieved by ...
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The Lov\'asz theta function $\theta(G)$ of a graph $G$ has attracted a lot of attention for its connection with diverse issues, such as communicating without errors and computing large cliques in graphs. Indeed this function enjoys the remarkable property of being computable in polynomial time, despite being sandwitched between clique ...
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Entanglement is an essential resource for quantum communication and quantum computation, similar to shared random bits in the classical world. Entanglement distillation extracts nearly-perfect entanglement from imperfect entangled state. The classical communication complexity of these protocols is the minimal amount of classical information that needs to be exchanged for the ...
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We study deterministic one-way communication complexity of functions with Hankel communication matrices. Some structural properties of such matrices are establishedand applied to the one-way two-party communication complexity of symmetric Boolean functions.It is shown that the number of required communication bits does not depend on ...
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We consider Total Functional $\NP$ ($\TFNP$) search problems. Such problems are based on combinatorial principles that guarantee, through locally checkable conditions, that a solution to the problem exists in an exponentially-large domain, and have the property that any solution has a polynomial-size witness that can be verified in polynomial time. ...
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We study the problem of non-interactive correlation distillation(NICD). Suppose Alice and Bob each has a string, denoted by$A=a_0a_1\cdots a_{n-1}$ and $B=b_0b_1\cdots b_{n-1}$,respectively. Furthermore, for every $k=0,1,...,n-1$, $(a_k,b_k)$ isindependently drawn from a distribution $\noise$, known as the ``noisemode''. Alice and Bob wish to ``distill'' the correlationmore >>>

Secure computation is one of the most fundamental cryptographic tasks.It is known that all functions can be computed securely in theinformation theoretic setting, given access to a black box for somecomplete function such as AND. However, without such a black box, notall functions can be securely ...
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We show that any 1-round 2-server Private InformationRetrieval Protocol where the answers are 1-bit long must ask questionsthat are at least $n-2$ bits long, which is nearly equal to the known$n-1$ upper bound. This improves upon the approximately $0.25n$ lowerbound of Kerenidis and de Wolf while ...
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